The Large Hadron Collider forward (LHCf) experiment is designed to use the LHC to verify the hadronic-interaction models used in cosmic-ray physics. Forward baryon production is one of the crucial points to understand the development of cosmic-ray showers. We report the neutron-energy spectra for LHC $\sqrt{s}$=7 TeV proton–proton collisions with the pseudo-rapidity η ranging from 8.81 to 8.99, from 8.99 to 9.22, and from 10.76 to infinity. The measured energy spectra obtained from the two independent calorimeters of Arm1 and Arm2 show the same characteristic feature before unfolding the detector responses. We unfolded the measured spectra by using the multidimensional unfolding method based on Bayesian theory, and the unfolded spectra were compared with current hadronic-interaction models. The QGSJET II-03 model predicts a high neutron production rate at the highest pseudo-rapidity range similar to our results, and the DPMJET 3.04 model describes our results well at the lower pseudo-rapidity ranges. However, no model perfectly explains the experimental results over the entire pseudo-rapidity range. The experimental data indicate a more abundant neutron production rate relative to the photon production than any model predictions studied here.